»,1.s 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/actuarialsocietyOOhendrich 


ACTUARIAL   SOCIETY 
EXAMINATIONS   IN    1905 

QUESTIONS    AND    SOLUTIONS 
REPRINTED  FROM  RECENT  ISSUES  OF 

THE  AIHERICAN  UdDEliWHiTEll 

AND 

THE    FUNDAMENTAL 
PRINCIPLES  OF  PROBABILITY 

BY 

Robert    Henderson,    B.  A.,  F.  I.  A.,  F.  A.  S. 


MCM  VI 

Thrift  Publishing  Company 
141   Bkoadwat,  New  Yokk  City 


^    OF  THE 


UNIVERSITY 

OF 


GEHtRI^^ 


Copyright,   1906 

BY    THE 

Thrift  Publishing  Company 


PREFACE 

J^R.  ROBERT  HENDERSON,  the  author  of 
the  solutions  of  Actuarial  Society  examination 
questions  of  1905,  presented  in  this  book,  and  of  the 
accompanying  essay  on  Probability,  was  born  May 
24th,  1 87 1,  at  Russell,  near  Ottawa,  Canada.  He 
was  graduated  from  Toronto  University,  with  the  de- 
gree of  B.  A.  in  1891,  and  was  a  University  Fellow  in 
mathematics  for  one  year.  In  1892,  he  went  into  the 
Dominion  Insurance  Department,  Ottawa,  Canada, 
under  the  Actuary  of  the  Department,  Mr.  A.  K. 
Blackadar,  F.  I.  A.,  F.  A.  S.  He  remained  until 
July,  1897,  when  he  became  associated  with  the  Actu- 
arial Department  of  the  Equitable  Life  Assurance  So- 
ciety of  the  United  States,  of  which  Society  he  was 
appointed  Assistant  Actuary  in  September,  1903. 

In  1896,  Mr.  Henderson  took  his  final  examination 
and  became  a  Fellow  of  the  Institute  of  Actuaries.  He 
was  enrolled  as  an  Associate  of  the  Actuarial  Society 
in  1900,  and  was  admitted  on  examination  as  a  Fellow 
in  1902.  By  appointment  of  the  President  of  the 
Actuarial  Society,  he  was  a  member  of  the  Committee 
on  Examination  for  the  years  1903,  1904  and  1905, 
the  last  year  as  Chairman.  At  the  annual  meeting  of 
the  Society,May  18,  1905,  he  was  elected  a  member  of 
the  Council  to  serve  for  three  years.  To  the  'Trans- 
actions of  the  Actuarial  Society,  his  principal  contribu- 
tions have  been,  "Frequency  Curves  and  Moments'*, 


167288 


May  19,  1904,  Vol.  VIII,  pp.  30-42;  and  "Note  on 
Limit  of  Risk",  May  18,  1905,  Vol.  IX,  pp.  40-46. 
Mr.  Henderson  has  given  the  solutions  of  the 
questions  in  the  examination  of  1905  of  the  Actuarial 
Society  in  a  clear,  concise  and  logical  manner.  His 
essay  on  "  The  Fundamental  Principles  of  Proba- 
bility" is  logical  rather  than  mathematical  in  charac- 
ter and  is  written  along  the  lines  of  the  most  recent 
treatises.  In  the  belief  that  these  solutions  and  this 
essay  will  be  interesting  and  useful  to  many  persons 
they  are  put  in  permanent  form  in  this  book.  The 
utility  of  the  book  is  enhanced  by  the  fact  that  the  left 
hand  pages  are  blank  for  the  convenience  of  those  who 
desire  to  make  additions,  criticisms  or  annotations  as  to 
the  particular  subjects  under  consideration. 

The  American  Underwriter 


CONTENTS 

PAGE 

Introduction  9 

Syllabus  of  Actuarial  Society  Examinations  15 

Questions  and  Solutions — 

I     Associate,  Section  A  iQ 

II  Associate,  Section  B  33 
III     Fellow  45 

The  Fundamental  Principles  of  Probability — 

I     The  Measurement  of  Probabilities  71 

II     The  Combination  of  Probabilities  75 

III  Expectation,  or  Mean  Values  85 

IV  Repeated  Trials  89 


UNIVERSITY 

OF 


INTRODUCTION 

rr^HE  following  solutions  of  the  questions  proposed  in  the 
•^  Actuarial  Society's  examination  of  1905  were  origin- 
ally prepared  for  publication  in  The  American  Underwriter 
and  they  are  now  brought  together  in  more  permanent  form  in 
the  hope  that  they  be  of  some  assistance  to  students  preparing 
for  the  examinations  of  the  Society.  In  this  connection  the 
author  considers  it  proper  to  point  out,  especially  in  view  ot 
his  connection  with  the  examination  committee  of  that  year, 
that  the  solutions  given  are  intended  only  as  specimens  and  not 
as  the  only  appropriate  answers  to  the  questions. 

Perhaps  a  few  words  as  to  the  principles  adopted  in  prepar- 
ing these  solutions  may  not  be  out  of  place.  Where  the  ques- 
tions referred  to  subjects  taken  up  in  text-books  which  were 
likely  to  be  in  the  hands  of  students  and  others  a  mere  refer- 
ence was  in  most  cases  considered  sufficient.  With  regard  to 
questions  in  algebra — in  particular — references  are  made  to  the 
text-book  in  which  the  author  could  most  readily  verify  the  ref- 
erences and  to  which  he  himself  most  frequently  refers.  The 
subjects  of  these  questions  will,  however,  be  found  taken  up  in 
any  standard  treatise  on  the  subject  and  the  student  will  refer 
to  the  particular  treatise  most  convenient  to  himself. 

In  certain  cases  where  an  alternative  treatment  of  a  subject 
taken  up  in  the  text-books  seemed  worthy  of  notice  it  has  been 
incorporated  in  the  solution. 


In  the  discussion  of  questions  of  a  general  nature  the  en- 
deavor has  been,  so  far  as  possible,  to  look  at  the  questions  from 
a  fresh  viewpoint.  In  all  cases  the  answers  have  been  made  as 
concise  as  possible  and  such  as  might  be  written  by  a  candidate 
working  under  a  time  limit. 

•  A  copy  is  included  of  the  syllabus  under  which  the  examina- 
tion was  held.  It  is  to  be  noted,  however,  that  a  change  has 
been  made  in  this  syllabus  which  is  intended  to  go  into  effect 
at  the  spring  examinations  in  1907.  This  change  consists  in 
adding  to  the  final  examination  the  subjects  of  Life  Insurance 
Bookkeeping,  Investments  and  Banking  and  Finance,  and  div- 
iding the  examination  into  two  sections  which  may  be  taken  in 
different  years. 

The  opportunity  is  taken  in  collecting  these  solutions  in  book 
form  to  append  a  brief  exposition  of  the  fundamental  principles 
of  the  theory  of  probability  which  was  suggested  by  the  two 
questions  on  that  subject  appearing  in  the  examination  papers. 
This  essay  is  intended  to  show  the  connection  of  the  mathemat- 
ical theory  of  the  subject  with  the  logical  theory  as  developed 
by  the  best  modern  writers  on  that  subject. 

Robert  Henderson 


11 


ACTUARIAL  SOCIETY 
EXAMINATIONS  IN   1905 


SYLLABUS  OF  EXAMINATIONS 

Associate 

Section  A 

1.  Arithmetic,  elementary  Algebra  and  the  principles  of 
double-entry  Bookkeeping. 

2.  The  following  subjects  in  advanced  Algebra: 

a.  Permutations   and   Combinations. 

b.  Binomial   Theorem. 

c.  Series. 

d.  Theory  and  use  of  Logarithms. 

e.  The   elements  of   Finite  Differences,   including  in- 
terpolation and  Summation. 

3.  The  elements  of  the  Theory  of  Probabilities. 

4.  Compound  Interest  and  Annuities  Certain. 

5.  Elementary  plane  Geometry.* 

6.  Practical  examples  in  the  foregoing  subjects. 
Section  B 

1.  The  application  of  the  Theory  of  Probabilities  to  Life 
Contingencies. 

2.  Theory  of  Annuities  and  Assurances,  including  the  theoiy 
and  use  of  Commutation  Tables  and  the  computation  of  prem- 
iums for  usual  contract. 

3.  Valuation  of  ordinary  forms  of  policies. 

4.  Practical  examples  on  all  the  above,  involving  Joint  as 
well  as  Single  lives. 


15 


5.  General  nature  of  Insurance  contracts. 

6.  The  outines  of  the  history  of  Life  Insurance. 

7.  The  source  and  characteristics  of  the  principal  Mortality^ 
Tables. 

Fellow 

1.  Methods  of  constructing  and  graduating  Mortality  Tables 
and  the  use  of  the  formulas  of  Gompertz  and  Makeham. 

2.  Methods  of  loading  premiums  to  provide  for  expenses  and 
contingencies. 

3.  Valuation  of  the  liabilities  and  assets  of  Life  Insurance 
companies. 

4.  The  assesment  of  expenses  and  the  distribution  of  surplus. 

5.  Practical   treatment  of  cases  of  alteration  or  surrender 
of  Life  Insurance  contracts. 

6.  Application  of  the  Calculus  of  Finite  Differences  and  of 
the  Differential  and  Integral  Calculus  to  Life  Contingencies. 

7.  Laws  of  the  United  States  and  Canada  relating  to  Life 
Insurance. 

8.  Insurance  of  Under-Average  Lives  and  extra  premiums 
for   Special  Hazards. 

*Note  for   Canadian  candidates.     Euclid  Books  I,  II  and  III  will  be  consid- 
ered  sufficient   for  this   subject. 


17 


ERRATA. 

Page   23,  Question     6,     line    4  —  Insert   8  after    word 

"computing-." 
Page   25,  Question  8,  line  2  —  Read   "annual  "  in  place 
of  "anual." 
Question  10,  line  4  —  read  numerator  "2401." 
Page   29,   Solution   13  (a),   line   4  —  Make   "j,-,    read  as 
index. 
Solution  13  (a),  line  5  —  Make  "  n"  curtate. 
Page  31,  Solution    14,    next    line    to    bottom  —  Insert 

"angle"  before  "P  AB\" 
Page   33,  Solution  1,  line  7  —  Read  "4  +  i"  in  place  of 

"/         1." 
Page   41,  Solution  10,  line  2,  —  Read  ".02186  X  ''  before 
the  fraction  in  place  of  ".02186  +•" 
Question  11,  line  1  —  Read  A\y^. 
Solution  12,  line  4  —  Read  "M./'  in  place  of 

"M.r." 

Page   43,  Solution  14  (a),  line  3  —  Read  "sixty"  in  place 
of  "twenty-three." 

Page   45,  Solution     1,     line    11   —  Insert    "by  "    after 

"  existing." 
Page   53,  Solution  11,  line  7  —  Read  "(1  +/)  "  in  place 

of   "(^+y)." 
Page   57,  Solution  15,  line   7  —  Read  "?/i"   in  place  of 

Page   61,  Question  22  (b),   line  2 --  Read  "4"  in  place 

of  "/^." 
Page   91,  Second  paragraph,  line  6  —  Read  "  ;/  -|-  1  "  in 

place  of  "  n  —  1." 


ACTUARIAL  SOCIETY  EXAMINATIONS 

Solutions  of  Questions  in  Examination  for  Admission  as  Asso- 
ciate OR  Fellow,  Held  April  13  and  14,   1905 

N.  B. — Actuarial  Society  or  American  notation 

Associate— Section  A 

V   I.  (a)    Simplify   ^  ~  ^  +  tit 

(d)  Extract  the  square  root  of  1,012,766,976. 

{a)  I. II. 

{b)  31,824. 

2.  (a)  Prove  that  a  quadratic  equation  can  have  only  two  roots, 

(b)  Solve  the  equation. 

^8  U  —  b)  jx  —  c)  jx  —  c)  (x  —  a)  ^ 

(a  —  b){a  —  c)'^        (b-  c)  {b  —  a) 

(a)  Let  a  and  ^  be  two  roots  of  the  equation,  then  since  x  —  a  and 
X  —  /J  are  factors  the  equation  must  take  the  form  a  {x  —  a) 
{x  —  ^)  =  o.  If,  then,  any  other  value :v  of  x  satisfies  this  equation 
we  have  a  {y  —  oi)  {y  —  /?)  =  o  which  is  impossible  unless  a  =  o/\n 
which  case  the  equation  would  vanish  identically.  Therefore,  the 
equation  cannot  have  more  than  two  roots. 

(b)  Substituting  a  for  x  in  the  equation  it  becomes 

a^  \--l    \--^\  ^o  =  a^ 
{a  —  b)    {a  —  c) 

Since  this  is  an  identity  a  satisfies  the  equation,  similarly  b 
satisfies  it  and  the  equation  does  not  vanish  identically  since  x  =  c 
does  not  satisfy  it.  As  the  equation  is  of  the  second  degree  and  does 
not  vanish  identically  it  has  only  two  roots  and  these  we  have  seen 
to  be  a  and  b. 


19 


/  3.  (a)  Given  that  the  Binominal  Theorem  is  true  when  the  index 
is  a  positive  integer,  prove  that  it  is  also  true  for  negative  or  frac- 
tional indices. 

(d)    Expand  (i  —  xT"^  . 
(a)     Eor  proof  of  first  part  of  question  refer  to  C.  Smith's  Algebra, 
2d  Edition,  Article  283. 
(d)    By  the  Binomial  Theorem. 

I  1.2  1.2.3 

+  etc. 
=  I   +  2  a;  +  3  ic2   +  4  :j;3   _f.  g^c. 

/    4.    Expand  e'^  in  ascending  powers  of  x. 

Hence,  where  Cq,  Ci,  etc.,  are  the  coefficients  of  the  various  powers 
of  x  in  (i  +  xf  prove  that  Cq  (a  +  w)*"  —  c\  (a   +  n  —  if^  -\-c* 
(a  +  n  —  2)*"  — etc.,  vanishes  if  m  is  less  than  n  and  is  equal  to  \n  if 
m  is  equal  to  n  {m  and  n  being  both  integral). 

e*  =  \  +^+-r-  +  -i ^-  etc. 

12  |3 

For  proof  refer  to  C.  Smith's  Algebra,  2d  Edition,  Article  302. 

^0  ia  +  n^  -c^{a  +  n  —  if  +  ^^  (a  +  «  —  2)^  —  etc.,  is  \m 
times  the  coefficient  of  x^  in  the  expansion  of  Cq  e  (a  +  «)*  _ 
^^^(a  +  «-l)^  +  ^2  ^(^  +  "-2^^- etc. 

or.  of^^^(^^-i)«. 

But  (^*  —  I)  =  i*;  +  1^  +  etc. 

.'.  (^*  _  i)«  =  ^«  +  ^  ^«  +  i  +  etc. 
^  2 

and  e^^  =  i  +  a  x  +  etc. 

.•.e^'={e''-ir  =  x-^  +  {a  +^)^«  +  i  +  etc. 

So  that  the  coefficient  of  all  powers  of  x  less  than  n  is  zero,  and 
that  of  x^  is  unity.     Hence  the  proposition  follows. 

5.  How  are  probabilities  measured?  Give  instances  where  addi- 
tion, subtraction,  multiplication  and  division  of  probabilities,  re- 
spectively, give  inter  pre  table  results,  stating  in  each  case  the  relation 
of  the  resulting  probability  to  those  operated  upon. 


21 


The  measure  of  the  probability  of  an  event  happening  under  given 
circumstances  is  the  limit  of  the  ratio  of  the  number  of  times  the 
event  happens  under  the  given  circumstances,  to  the  number  of 
times  the  given  circumstances  occur,  when  the  number  of  cases 
is  indefinitely  increased.  The  probabilities  of  two  mutually  exclu- 
sive events  may  be  added  together,  the  result  giving  the  probability 
that  either  one  or  the  other  will  happen.  If  two  events  be  so  related 
that  the  first  cannot  happen  unless  the  second  does,  the  probability 
of  the  first  may  be  subtracted  from  that  of  the  second,  the  result 
giving  the  probability  that  the  second  will  happen  unaccompanied  by 
the  first.  The  probability  of  an  event  may  be  multiplied  by  the 
chance  that  if  it  happens  a  second  event  will  happen  also,  the  result 
being  the  probability  that  both  will  happen.  Conversely  the  proba- 
bility of  a  compound  event  may  be  divided  by  the  probability  of  one 
of  its  components,  the  result  being  the  probability  that  if  that  event 
happens  the  other  will  happen  also. 

6.  //  i  is  the  effective  annual  rate  of  interest,  j  the  nominal  rate 
convertible  m  times  a  year,  d  the  rate  of  discount  and  8  the  force  of 
interest;  express  each  in  a  series  of  ascending  powers  of  each  of  the 
others.  Give  an  approximate  method  of  computing  d* for  a  given 
rate  of  interest. 

6.  If  i  is  the  effective  annual  rate  of  interest,  /  the  nominal  rate 
convertible  m  times  a  year,  d  the  rate  of  discount  and  S  the  force  of 
interest ;  express  each  in  a  series  of  ascending  powers  of  each  of  the 
others.  Give  an  approximate  method  of  computing  d  for  a  gi-en 
rate  of  interest. 

See  Institute  of  Actuaries'  Text  Book,  Part  i,  new  edition,  Chap- 
ter I,  Article  30, 

From  the  expressions  for  i  and  d  in  terms  of  8  we  have 

i  +  d  =  26  -\ h  etc. 

or  approximately  d  =  — - — 

also  id=  6'  +  —  5*  +  etc. 

so  that  approximately  S  =    V  i  d. 

7.  Having  given  the  value  of  an  annuity  certain  for  a  term  of 
years,  deduce  a  formula  for  fiiiding  the  approximate  rate  of  interest. 

Suppose  a  is  the  value  of  the  annuity  and  «  the  period,  and  let  i 
be  the  rate  of  interest.     Let  also  a^,  near  to  a,  be  the  value  at  some 

1  1  •       .u  1     «  —  ^1         d  a^ 

known  rate  Zi,  then  we  have  approximately 


di-i 


but  ix  «!  =  I  —  (I  +  ^i) 

.-.ai  ^i^^=  «(i  +  z\)-"  +  ^=  nv^^"^' 

.  «+  I 

a  a,  ai  —  n  Vi 

or  -r^  =  —  — 

^H  i^ 

«i  —  ^  .        . 

whence  i  —  t^  =  t  ^  -~~  or  i  =  t^  +  t\~ 

a,  —  n  v^^  ~^^  a, 

23 


/0 

(  UNI 


^     OF  The 

UNlVERSi 

OF 


y 


/ 


If  desired  this  rate  may  be  used  as  a  basis  for  a  further  approxi- 
mation. 

8.    Define  Revenue  Account,  Balance  Sheet  and  Inventory. 

What  items  in  the  anual  statement  of  a  life  insurance  company 
are  ordinarily  the  result  of  an  inventory? 

The  Revenue  Account  shows  on  one  side  the  amount  of  the  policy 
reserve  and  the  surplus  at  the  beginning  of  the  year  and  the  income 
earned  during  the  year,  and  on  the  other  side  the  expenditure  in- 
curred during  the  year,  the  balance  being  the  amount  of  the  policy 
reserve  and  surplus  at  the  end  of  the  year. 

The  Balance  Sheet  is  made  up  of  the  balance  of  the  other  ac- 
counts showing,  on  one  side,  the  capital  stock,  if  any,  the  policy 
reserve,  the  liabilities  outstanding  on  the  various  accounts  and  the 
surplus,  and  on  the  other  side  the  assets  on  hand 

An  Inventory  is  a  computation  of  the  total  amount  of  a  given  kind 
of  assets  or  liabilities  made  by  valuing  the  individual  items.  Interest 
and  rents  due  and  accrued  and  market  value  of  bonds  and  stocks 
over  book  value  are  ordinarily  determined  by  inventory.  The  lia- 
bility for  reserve  may  also  be  considered  as  so  determined. 

*-    9.     Fhid  the    value  of  ^C^,    the    number    of  combinations  of  n 
different  things  taken  r  at  a  time. 

Prove  that,  if  x  and  y  be  any  two  positive  integers,  then  will 
x-4-yC„  =  ^C„   +  x^n-^^  y^ ^     +  «C„_,  yC^    + +  yC„. 

See  C.  Smith's  Algebra,  2d  Edition,  Articles  244  and  248. 
^      10.     Given  log^o2  =   .30103  afid  log^oS  =   .47712,  calculate  th€ 
logarithms  of  all  numbers  from  4  to   10  inclusive,   using,   where 
necessary,  a  finite  difference  formula. 
From  these  values  calculate  log^  „  f  lU  • 
We  have 

log   4  =  2  log  2  =  .60206 

Jog    5  =  log  10  —  log  2  =  I  —  .30103  =  .69897 

log   6  =  log  2  -I-  log  3  =  .77815 

log    8=3  log  2  =  .90309 

jog    9  =  2  log  3  =  .95424 

log  10  =  1. 00000 

To  determine  log  7  we  have 

log  45  =  log  5  +  log   9  =  1. 65321. 

log  48  =  log6  +  log    8  =  1. 68124 

log  50  =  log  5  +  log  10  =  1.69897  '^ 

log  54  =  log6  +  log    9  =  1.73239 
Whence  interpolating  by  Lagrange's  formula- 
log  49=  _  Vlog45  +  flog  48  4^  ^  log  56 -sV  log  54  =  1.69020  I 

log    7  =  i  log  49  =  .84510  \ 

Also  we  have  log  \\^\  =  4  log  7  _  log  3  —  3  log  2  —  2  =  .00019  ^ 

11.     (a)     Define  central  differences. 

Kb)     Expand     « + «         x-n    and      x^n        x  —  n  jn   terms  of  u 
22  * 

and  its  central  differences . 


25 


(a)  Central  differences  are  the  differences  of  a  function  re- 
ferred to  the  central  value  of  the  variable.  As  there  is  no  value 
of  the  odd  orders  of  differences  corresponding  to  the  values  of  the 
original  function  the  mean  of  the  two  adjacent  values  is  used. 

(d)    We  have, 
u    ,        =  u      +  na    +   n^  .      ,    n  {n*  —  i)  ,     «'  («*  —  i) 

'+"       '       '     K  '    ~]i —  "'     ~\1 — 

dx  +  etc. 

x  —  n  X  *|2  |_3_  * 

-      «'('''  -'">        d^  +  etc. 


I_4_ 

where  a^,  d^,  c^,  d^,   etc    are  the    successive   central    differences 
corresponding    to  u^ 

Whence  "'+>•  +  "-"  =  «,+  £!«,  +  "'  '"•  -  "  d^  +  etc. 

2  *  J2        "^  l_4_ 

2  /J_ 

12.  (fl)  r/i^  probability  of  the  happening  of  a  certain  event  is 
p,  and  the  probability  of  its  failure  is  q;  what  is  the  probability  of 
its  happening  exactly  r  times  in  n  trials? 

(b)  for  what  value  of  ris  this  probability  the  greatest? 

(c)  Does  this  maximum  probability  increase  or  decrease  as  n 
increases ? 

(a)  The  chance  of  the  event  happening  on  any  particular  r  trials 
and  failing  on  the  remainder  n  —  r  is  ^''  q"~^-     But  the  number  of 


% 


different   combinations   of  r  trials   is  -; ; so   that   the   total 

I  r  \n  —  r 

\n_ 

probability  is ; >, »-  n^—^ 

I  r   \n  —  r  ^     ^ 

(b)    This  value  is  the  greatest  for  the  largest  value  of  r  for  which 

\n  \n  ^  ,  , 

I  r  \n  —  r  \  r—  i  \n  —  r  4-  i 

{n  — r  +  i)  p>  rq,  or  r{p  ■\-  q)<{n  ■\-  i)  p 

or,  since  p  ■}-  q  =  i,  when  r  is  the  integral  part  of  (n  +  i)  p. 


27 


(c)    Putting  now  («  +  i)  for  n  the  maximum  probability  is  either 

|«  -f  I  .  ,  \n  +  I  ■ 

P''  g"-^  +  ^or    .      .    _    ,  p^'+^g*'-' the 


\r   \n  —  r  +  I  |  r  -»-  i    |  n  —  r 

ratio  of  which  to  the  preceding  is  either  ^  or  -^ -^ 

n  —  r  +  I  r  +  1 

both  of  which  are  less  than  unity  since  («  +  i)  P  lies  between  r  and 

r  -\-  I  and  consequently  {n  -\-  i)  q  between  n  —  r  and  n  —  r  -f-  i. 

/  13.     (o)     What  is  the  value  of  an  annuity  certain  for  n  years, 

payable  p  times  per  annum  at  the  nominal  rate  j  convertible  m  times 
a  year. 

^  (b)  An  annual  annuity  running  n  years  is  worth  twenty-five 
year's  purchase;  one  running  2n  years  is  worth  thirty  years'  pur- 
chase.    What  is  the  rate  of  interest f 

(a)  We  have  where  i  is  the  effective  rate  of  interest  per  annum  ; 
(p)       I  .  X         i  ^^  I  —  e/» 


"'    ^  /[(i  +  O'-iJ 

but  I  +  z  =  f  I  +  -^  \    ,  so  that 


'[(■*i)7-J 


(w)         I  —  Z/*" 

Where/  is  equal  to  m,  this  becomes  a  — -  = 


n  J 


(b)  U  a  — \=  — : —  =  25 

u\  J 


and 

2MJ 

=  ^- 

y 

^n 

=  30, 

a 

have  - 

2«| 

I  — 

^^n 

I 

+ 

2/" 

6 

(//O 

I  - 

-  2/" 

5 

n 


4  .  C^') 

therefore  i  —  z/«  =  -  =  25/,         since  rt  —  =  25 

therefore/  =  -^,  or  the  nominal  rate  of  interest,  convertible  with  the  same 
frequency  as  the  annuity  is  payable,  is  3^  per  cent. 


29 


y  lA.  On  the  side  A  B,  produced  if  necessary^  of  a  triangle 
A  B  C,  A  C  is  taken  equal  to  A  C]  similarly  <?«  A  C,  A  B  is  taken 
tqual  to  A  B,  and  the  line  B^  C  is  drawn  to  cut  B  C  in  P.  Prove 
that  the  line    A  P  bisects  the  angle  BAG. 

A 


Let  the  construction  be  made  as  described  and  suppose 
C    is  ia  A  B  produced  and  consequently  B    in  A  C. 

Then  in  the  triangles  A  B  C  and  A  B  ^C^  since  A  B=A  ^' 
A  C=  A  C^  and  angle  B  A  C= angle  ^'  A  c/  the  triangles 
are  equal  in  every  respect,  or  angle  A  B  C  wangle  ABC, 
and  angle   A   C  B=angle   A  c/   B'' 

Also  since  angles  CT  B  P  and  C  B  P  are  supplementary 
to   the  angles  ABC  and   ABC      they  are  equal. 

Also  since  A  B  =  A  B  and  A  C  =  A  C  therefore 
B    C^  =    B^  C. 

Then  in  the  triangles  P  B  C^and  P  B^  C,  B  C^  =  B'^  C 
angle  P  B  C^=  angle  P  B^  C  and  angle  P  d  B-=angIe  P  C  B/ 
therefore   P   B  =  P  B^ 

Then  in  the  triangles  P  A  B  and  P  A  b{  P  B  -=  P  B^ 
A  B  =  A  B^  *nd  P  A  is  common  therefore  angle  P  A  B  «.  P  A  B^ 
or  P  A  bisects  the  angle  BAG. 


31 


Associate— Section  B 

y       1.     What  does  the  mortality  table  represent,  and  of  what  columns 
does  it  usually  consist? 
Express  in  terms  of  the  mortality  table  the  probabilities: 

(a)  That  one  of  the  lives  (x)  and  (y),  will  survive  n  years  and 
the  other  will  fail  within  n  years. 

(b)  That  at  least  one  of  the  lives  will  fail  in  n  years. 
(c)l  That  both  will  survive  n  years. 

(d)  That  both  will  die  within  n  years. 

(e)  That  the  first  death  will  happen  in  the  n^h  year  from  the 
present  time. 

It  represents  the  history  as  regards  mortality  of  a  representative 
group  of  people,  and  indicates  the  proportion  which  dies  in  each  year 
of  age  and  the  proportion  which  survives  to  the  end  of  each  year. 
The  mortality  table  proper  consists  of  the  column  of  /^  showing  the 
proportion  of  those  attaining  the  initial  age  which  survives  to  each  sub- 
sequent age,  and  ths  column  of  d^  showing  the  proportion  of  the  same 
group  which  dies  in  each  year,  so  that  necessarily  d^  =/^  — /    ^+l 

(a)      ^p^(i-,Py)  "r  ^Py{i-^p^) 

=  nPx  +  nPy  —  ^nPx  •  nPy 

^  x-\-n        ^  y+n  ^x-\-n:  ^y+n 

°      I,      ^       ly      ~^  '«•', 

^  r-^n  :  ^y-\-n 

(*>      ^-nPx:nPy=^   ^- 1 } 

*«  •  ''y 

^x+n  :    ^  y-^fi 


<^)     nPx:nPy=  ^        ^ 


ilx—  ^x-f  «^  ^^y-  ^y+n^ 


''X  :   ''y 

^x  +  ti         ^y-\-n         ^x±n^Jy±n 

=  I  —  — + ; ; 

l^  ly  Ix  :    ^ 

(^)     n-lPx.n-lPy  —  ftPx-  nPy 

_    ^x  +  n  —  l-^y-\-n  —  l  ~  ^  x+  n  :  ^  y -{- n 


33 


2.  Express  the  value  of  a^  m  terms  of  the  mortaltty  table. 
Prove  that  a^  <  ar^\ . 

If  annuities  of  a  unit  be  issued  on  each  of  l^  lives  aged  x  the  total 
amount  to  be  paid  at  the  end  of  the  first  year  is  /<p^i,  at  the  end  the 
second  year  l^j^^y  and  so  on.    So  that  equating  present  values  we  have 
/;,  fl^  =  z/  /,^_i  +  z/^  /^  +  2  +  etc. 

t'  /-  .  1  +  z/^  ^x  +  2  +  etc. 
or  a  ^  =  — t-L± 

I, 

We  know  that  A^  >  z/  ^*  For  the  former  is  the  arithmetic 

and  the  latter  is  the  geometric  mean  of  the  present  values  of  the 
death  claims.     Therefore 

I  —  (I  +  /)  A_       I  _  z,  ^*      . 
a^  =  -, ?  <  i ^ —  ;  that  is  <  a^-j. 

3.  On  De  Moivre's  hypothesis  find  the  value  of  a  life  annuity  in 
terms  of  the  expectation  of  life. 

Does  this  formula  give  results  sufficiently  close  for  practical  useT 
On  De  Moivre's  hypothesis,  if  n  is  the  complement  of  life,  we 

o         n 
have,  since  the  decrements  are  supposed  uniform,  ^x~~o 

Also,  A,  =  -^  so  that 

I  -  A^      n-a~\        2^^  — a,/^| 


d  net  ^1      ^ 

Owing  to  the  lack  of  conformity  of  mortality  tables  with 
De  Moivre's  hypothesis  this  formula  does  not  give  results  suffi- 
ciently close  for  practical  purposes. 

4.  Deduce  formulas  for  the  net  single  and  annual  premiums  for  a 
survivorship  annuity  payable  to  (x)  after  the  death  of  (y). 

The  payment  under  the  survivorship  annuity  will  be  due  at  the  end 
of  any  year  if  (x)  be  alive  and  (y)  be  dead.  The  net  single  premium 
for  the  benefit  is  therefore, 

-    ^^''uPx-^^""  nP.    ■   nPy 

=  a    —  a, 

X  X  y 

The  annual  premium  is  ordinarily  payable  during  the  joint  lives  so 
that  denoting  its  value  by  n  we  have 

or  ?r  =  = —  I . 


35 


5.  Having  given  the  values  of  D^,  N^  and  S^,  show  how  the 
values  of  C^,  M^  and  Rj.    may  be  ascertained. 

Find  the  annual  premium  for  whole  life  insurance,  all  premiums 
paid  to  be  returned  at  death,  assuming  that  the  premium  is  loaded  by 
a  percentage  and  a  constant. 


=  D.-d  N. 


M.  =  ^C,  =  7;:S  D.-2D,^,  =  t,N^-N,+ 


X 


Let  TTj  =  office  premium  required  and  tc  =  net  premium  where 

7t^={l    -\-    k)7C   +   c 

Then  ;r  N^  =  AI^  +  R^l  (I  +  k)7C  +c^ 

or  TT  I  N^  -  (I  +  ^;  R^  I  =^  M^  +  c  R, 
M.  +  c  R,       - 

^        .r  a.  fc^  ^  ^          (I  +  ^)  M,  +  c  N,      ^'  +  ^^  n;   +  ^ 
''^i  =  (I  +  k)  n  +  c  = = — 

N,-(i+fe)R,  •    f,^u,^^ 

(I   +  ^)  P^  +   c 


K 

i-(i  +^)j^ 


6.  Investigate  the  relation  between  the  rates  of  mortality  in  two 
tables  giving  the  same  net  premium  reserves  on  ordinary  life  policies. 
What  conclusion  would  you  draw  regarding  the  effect  on  such  re- 
serves of  an  increase  in  the  rate  of  interest? 

We  have  for  all  values  of  n 

|n-iV^+P^    [(I     +    V)  =   P^^n_,   „  V,    +^^  +  „_,- 

n^  X  +^;.  +  n-i(l-nV^). 
Suppose  any  other  table  of  mortality  gives  the  same  terminal  re- 
serve values  and  denote  net  premiums  and  rates  of  mortality  according 
to  this  second  table  by  accented  letters  then  we  have 

]n-:V,+Pi    [(I    +^-)=„V,+9i  +  „_,(l-„V,). 

Subtracting  we  get  (P^- P^)  (i  4- *)  =  (5i  +  „_  ,  -  ^i^  „  _  j) 
or  Pi-PJ(i  +i)(i  +a^)  =  (5i^„_,-g,  +  „_,) 

(I     +«;c  +  n). 


37 


for  all  values  of  n  so  that 

(^i  —  g^)  (i  +  ax  +  i)  must  be  a  constant 

k 
or  q^.  can  be  expressed  in  the  form  q^  +  — T — 

X  -f-  I 

An  increase  in  the  rate  of  interest  is  equivalent  so  far  as  annuity- 
values  are  concerned  to  a  proportionate  reduction  of  the  values 
of  pj.  or,  for  adult  lives,  a  decreasing  addition  to  the  value  of  q^ 
whereas  in  order  to  give  the  same  reserves  an  increasing  addition 
to  q^  is  required,  consequently  we  should  expect  that  an  increase  in 
the  rate  of  interest  would  in  general  lower  the  reserves. 


7.  State  the  leading  points  of  difference  between  policies  now 
issued  in  America  and  those  issued  fifty  years  ago. 

Fifty  years  ago  policies  were  issued  on  comparatively  few  forms, 
principally  ordinary  life,  the  many  special  forms  which  are  now  such 
a  feature  of  the  business  being  then  unknown.  The  many  privileges 
and  options  as  to  surrender  values,  dividends,  etc.,  now  granted, 
were  then  absent  from  the  contract  which  on  the  other  hand  con- 
tained many  restrictions  and  prohibitions  relating  to  travel,  occupa- 
tion, and  residence.  Deferred  dividend  policies  were  first  issued  in 
1S68. 

8.  Fmd  the  values  ofQly]  Q  J. .  y^ff)  and  e^  .  y^) . 

Among  those  cases  where  (x)  actually  survives  (y),  what  is  the 
average  duration  of  the  period  of  survival  f 

See  Institute  of  Actuaries'  Text  Book,  Part  2,  Chapter  4,  Articles 
3,  13  and  16. 

In  a  large  number  N  of  cases  the  number  where  (a:)  survives  {y) 

is  N  QJ.    and  the  total  number  of  years  included  in  all  the  periods  of 
such  survival  is  N  e^ , ,  =   N  {e^  —  e^^.      Therefore,  the  average 

penod  IS  — p— . 


9.  Frotn  general  reasoning  find  the  value  of  A^  i?i  terms  of  a^. 

Express  also  its  value  in  terms  of  the  mortality  table  and  in  com- 
mutation symbols. 

See  Institute  of  Actuaries'  Text  Book,  Part  II,  Chapter  VII,  Arti- 
cles 32,  33.  41,  42,  43  and  44. 

10.  Give  formulas  for  the  ordinary  life,  limited  payment  life  and 
endoii'ment  assurance  annual  premium  rates  in  terms  of  annuity 
values  and  also  in  terms  'jf  commutation  columns. 

Given  P^.  =  .02186,  P^-|  =  .0^20  and  i  =  .03;  find  ^ ^. 

See  Institute  of  Actuaries'  Text  Book,  Part  II,  Chapter  VII,  Arti- 
cles 61,  69,  73,  79  and  80. 


30 


OF  THE 


UNIVERSITY 


We  have  „P,  = ^::=^  =  A,  (P  "j  -^  d)  = (P^j  +rf) 

I   +  ax:n—l\  ^x  '^  "■ 

(I  +  i)  P-1   +  «                          1-03   X  .04220  +  .03 
=  P =    .02186  % TT- 

^    (I  +  /)  P^  +  «  '^  1.03   X  .02186  +  .03 

.0734660 

=  .02186  X  1  =  .03058 

.0525158 

11.  Find  an  expression  for   A  '        and  show  how  to  express  in 

terms  of  simpler  functions  the  values  of  A  ^        A  ^^^  and  AJ  — 

See  Institute  of  Actuaries  Text  Book,  Part  II,  Chapter  XIII,  Ar- 
ticles 24,  26  and  27. 

12.  Write  down  formulae  for  the  value  of  a  limited  payment  life 
policy  by  the  prospective  method  and  by  the  retrospective  method, 
explaining  each.  State  under  what  conditions  they  give  identical 
results. 

Prospectively  „V  = =; where  m  is. 

the  premium  period. 

•     ,      w       '^.(Nx-N:,  +  „)-(M,-M,+,) 
Retrospectively  „V  = — 

The  difference  between  these  two  values  is ■ 

Dx+« 

which    vanishes  if  7t^  =   -tt — >^ ,  that  is,  if  the  premium 

X  X  -f  m 

valued  is  the  net  premium  for  the  policy  on  the  basis  of  valuation. 
Otherwise,  the  two  will  give  different  results,  unless  the  retrospec- 
tive formula  be  properly  modified. 

13.  State  as  briefly  as  possible  the  main  points  to  be  covered  in 
drawing  up  a  policy  of  insurance  on  the  limitedpayment  life  plan. 

The  special  points  to  be  covered  in  drafting  a  limited-payment  life 
polic3%  as  distinguished  from  other  forms,  are  that  the  face  of  the 
policy  is  payable  on  receipt  of  satisfactory  proofs  of  death  of  the 
assured  and  that  the  assurance  is  conditional  on  the  payment  of  the 
specified  premium  annually  in  advance  during  a  specified  period  or 
until  the  prior  death  of  the  assured.  These  are  the  distinguishing 
features  of  a  limited  payment  life  policy,  the  other  privileges  and 
conditions  being  in  general  common  to  this  and  other  forms.  The 
conditions  embody  such  restrictions  and  limitations   regarding  oc- 


41 


cupation,  travel,  residence,  suicide,  and  proof  of  age,  as  may  be 
considered  advisable,  and  usually  incorporate  by  reference  the  ap- 
plication as  part  of  the  contract.  The  privileges  refer  to  such 
features  as  incontestability,  grace  in  payment  of  premiums,  rein- 
statement, change  of  beneficiary,  loans  and  surrender  values. 

14.  (a)  What  are  the  leading  features  which  distinguish  the  O^ 
table  from  the  Actuaries'  table? 

(b)  What  was  found  to  be  the  experience  of  policies  with  and 
without  profits  respectively  and  of  endowments  as  compared  with 
whole-life  policies f 

(o)  As  regards  source  the  O^  table  is  distinguished  by  being 
the  result  of  the  recent  experience  of.  a  large  and  homogeneous  class 
of  risks,  being  the  experience  of  twonfirfthrcc  British  offices  during 
the  years  1863  to  1893  on  male  lives  assured  under  ordinary  life 
policies,  while  the  Actuaries'  table  is  based  on  the  whole  experience 
of  seventeen  offices  up  to  1839.  As  regards  methods,  the  O**  table 
was  made  up  on  lives,  while  the  Actuaries'  table  was  made  up  on 
policies,  except  that  in  some  companies  duplicates  in  the  same 
company  were  eliminated.  As  regards  results,  the  O^  table  shows  a 
relatively  very  low  rate  of  mortality  at  the  younger  ages  as  com- 
pared with  the  Actuaries'  table.  The  difference  in  rates  of  mortality 
becomes  relatively  very  small  as  the  age  advances.  The  expectation 
by  the  O^  table  is  about  five  per  cent,  higher  at  age  20  than  by  the 
Actuaries  table.  At  age  35  this  difference  is  reduced  to  about  two 
and  one  half  per  cent,  and  remains  at  that  figure  beyond  age  70. 

(b)  In  the  British  offices  experience,  the  mortality  on  lives  as- 
sured with  profits  was  much  lighter  than  on  those  assured  without 
profits,  and  the  experience  on  endowments  was  much  more  favor- 
able than  on  whole  life  policies. 


43 


Fellow- 


I.  Draft  a  memorandum  of  instructions  to  be  followed  by 
clerks  in  taking  out  the  mortality  experience  of  a  company,  the  result 
to  be  exhibited  in  the  form  of  select  tables. 

From  proper  record  enter  on  cards  the  age  at  entry,  date  of  issue 
and  date  and  mode  of  termination  of  risk  (if  terminated).  Include 
only  terminations  occurring  before  the  anniversary  in  the  year  1904. 
In  the  case  of  terminations  otherwise  than  by  death,  record,  as  the 
duration,  the  difference  between  the  calendar  years  of  entry  and 
exit.  In  the  case  of  deaths  add  one  year  to  the  number  of  com- 
plete policy  years  which  have  elapsed  between  date  of  issue  and 
date  of  death.  Take  the  age  at  entry  at  nearest  birthday.  Sort  cards 
into  deaths,  other  terminations  (withdrawals)  and  existing,  then 
sort  each  lot  by  age  at  entry.  Sort  the  deaths  and  withdrawals  at 
each  age  according  to  duration,  and  the  existin^calendar  year  of 
issue  (the  duration  being  the  difference  between  that  year  and  the 
year  of  termination  of  observations).  Record  on  a  schedule  for  each 
age  at  entry  the  total  number  for  each  duration,  separately  for  the 
deaths,  withdrawals  and  existing.  (The  cards  should  contain  further 
information  such  as  kind  of  policy,  residence,  etc.,  if  required  for  the 
purposes  of  subdivision). 


2.  On  the  assumption  of  Makeham's  law,  find  the  value  of  A^^  in 
terms  of  the  constants  and  A^y  Hence  give  approximately  the 
value  of  A  J.y  in  terms  of  A ^^,. 

We  have  by  Makeham's  law, 


/' 


+  t 


A  +  Bc*  +  * 
and>Uy^^=  A  +   B(7^  +  ^ 

where  A  =  —  log^  s. 
.-.  c^i^^j^t  +  log^s)  =  c'i^y^t  +  log^s) 
multiplying  then  through  by 
v^  f  p^y  and  integrating  with  respect  to  i,  we  get 

ovcyAiy  =  c'A^;  +  {c^-cy)iog^s'^^y 

or  ic'  +  c:>')Ai,,  =  c^' A^^  +  (c' -  cy)  log^  s'a ^  ^ 
=  c-A^y  +{c^-cy)^fll^ii-A^y) 


45 


But  a;^  =  yAi^and  A3.3,  =  -^  A ^^ approximately. 
Hence  substituting  and  reducing  we  have 


3.  ff'i^a/  various  forms  of  investment  are  now  open  to  life  insur- 
ance companies  ?    Discuss  their  relative  desirability . 

Railroad  Bonds :  They  yield  a  fair  return,  are  available  in  con- 
siderable quantity,  when  well  selected  furnish  ample  security,  and 
are  readily  convertible. 

Government  and  Mufiicipal  Bonds :  The  interest  return,  except 
for  small  issues,  is  comparatively  low,  but  they  are  sometimes  useful 
for  deposit  purposes. 

Mortgage  Loans  on  improved  property,  with  a  good  margin,  are  a 
safe  investment  and  yield  a  return  somewhat  higher  than  do  first- 
class  bonds. 

Stocks :  Great  care  in  selection  is  necessary.  Some  first-class 
railroad,  bank  and  trust  companies  stocks  however,  furnish  a  safe 
and  remunerative  investment. 

Loans  on  Policies :  This  is  one  of  the  most  desirable  forms  of 
investment,  but  only  a  limited  proportion  of  the  funds  can  be  invested 
in  this  way. 

4.  Show  how  the  reserve  on  any  policy  at  the  end  of  n  years  can 
be  expressed  in  terms  of  the  net  premium,  and  the  commutation 
symbols  for  age  at  entry  a7id  age  attained.  Hence  show  how  policies 
of  all  kifids  a7id  durations  may  be  grouped  together  by  attained  age 
for  valuation  purposes. 

Let  „V  be  the  reserve  at  the  end  of  n  years.  S  the  sum  assured 
and  re  the  net  premium,  then  we  have 

N^-N^  +  n         ^      M^-M^+n 

or„V=7r-— S. 

^x^n  ^x-\-n 

This  expresses  the  reserve  in  terms  of  the  net  premium  and  com- 
mutation symbols  for  age  at  entry  and  age  attained.  This  equation 
can  also  be  expressed  in  the  form 

nV=  S.A^_^„-  ;r  (I  4-  a^^J  + 

^x  +  n 


47 


D, 

=  S.  A,^^-  «-(!   +  a,^J    +6.- 

TT  N,  —  S.  M, 
where  fl  =  — and  z  is  some  convenient  advanced  age. 

z 

If  then  the  quantities  S.  ic  and  9  be  recorded  once  for  all  on  each 
policy  a  classification  may  be  made  by  age  attained  and  an  account 
kept  of  the  totals  of  the  three  functions  for  each  group.  These  can 
be  multiplied  by  the  respective  valuation  functions  A^^^^,  i  +  ^x-\-n 

and  — and  a  summation  will  give  the  valuation  required.     If  the 

^x+n 

conditions  of  the  policy  are  such  that  the  value  of  S  or  of  tt  changes 
after  the  policy  is  in  force,  a  corresponding  change  in  B  will  have  to 
be  made;  the  new  value  of  6  being  such  that  at  the  date  when  the 
change  takes  effect  the  value  on  the  new  set  of  constants  will  be  the 
same  as  on  the  old. 

5,  Under  what  conditions  is  the  declaration  of  a  simple  rever- 
sionary bonus  of  a  uniform  percentage  of  the  face  of  the  policy  for  all 
ages,  durations,  and  kinds  of  policy ,  equitable  f 

Profits  may  for  practical  purposes  be  divided  into  two  parts,  one  to 
be  apportioned  in  proportion  to  the  reserve  on  the  policy  and  the 
other  in  proportion  to  the  effective  loading,  so  that  the  equitable 
share  of  a  policy  may  be  expressed  in  the  form 


al-\-b.y=^al-\-b 


I -A, 


for  ordinary  life  policies ;  for  endowments  the  corresponding 
changes  in  the  formula  should  be  made.  If  the  proposed  method 
is  to  be  fair,  this  should  be  capable  of  expression  in  the  form 
^  A^_l_„  where  c  is  constant  for  all    values    of   x   and   n  that  is 

al-b  ^  _  ^     +   ^  __  ^    .  A,^„  =  c  A,  +  „  for  all  values  of  x 

^x  b       ^x 

and  n.     Hence  a  I  —  b =  o  ox  I  =  —  .  —r  or  the  effective 

I  —  A^  ad 

b 
loading  is  a  constant  percentage  of  the  net  premium,  also  *    =  c. 

I         Ay 

This  last  condition  cannot  be  fulfilled  as  A^  is  a  variable  and  b  and  c 

are  constants  unless  the  valuation  basis  is  varied  according  to  age 
at  issue  and  kind  of  policy  so  that  b  would  vary  also,  being  made 
proportional  to  the  difference  between  the  rate  of  interest  assumed 
and  that  actually  earned. 


49 


6.  Expand  -7 in  ascending  powers  of  x. 

(I  \  xY  —  \ 

Hence  derive  an  approximate  formula  for  the  sunt  of  a  series  when 

only  every  «<^  value  is  k7iown . 

See  Institute  of  Actuaries'  Text  Book,  Part  2,  Chapter  22,  Article 
30,  and  Chapter  24,  Article  20,  and  Transactions  of  Actuarial  Society 
of  America,  Volume  8,  page  58. 

7.  A  policy  is  drawn  payable  as  follows  :  "  unto  Jane  Doe,  bene- 
ficiary, wife  of  the  insured,''  without  any  further  provision.  The 
beneficiary  dies  and  the  insured  wishes  to  surrender  the  policy. 
Discuss  the  rights  of  the  insured  in  such  a  case. 

A  policy  so  worded  would  at  the  death  of  a  wife  inure  to  the  benefit 
of  the  children,  if  any,  and  according  to  the  law  of  some  states  could 
not  be  assigned.  Where  an  assignment  is  possible  the  children 
should  join  in  the  release,  either  individually,  if  adult,  or  if  minors, 
through  their  general  or  special  guardian.  Where  no  children  sur- 
vive, her  interest  will  pass  to  her  heirs. 

8.  A  set  of  select  tables  are  to  be  graduated  in  which  the  mor- 
tality in  the  early  policy  years  at  ages  belozv  twenty-five  at  entry  is 
higher  in  the  ungraduated  data  than  at  several  higher  ages.  What 
method  would  be  most  satisfactory?  Give  details  of  the  work  of 
applying  the  method  you  would  adopt. 

As  it  is  impossible  to  represent,  by  a  table  graduated  by  Make- 
ham's  law  unmodified  the  special  feature  referred  to,  that  law  should 
not  be  used  unless  an  extended  use  is  expected  to  be  made  of  the 
resulting  table  in  joint  life  calculations.  No  perfectly  satisfactory 
method  has,  however,  yet  been  devised  of  applying  summation  grad- 
uations, such  as  those  of  Woolhouse  and  Higham,  to  select  tables. 
The  graphic  method  of  graduation  would,  therefore,  be  the  most 
suitable.  The  expectations  of  life  for  the  separate  ages  at  entry 
should  be  made  from  the  ungraduated  data.  These  will  indicate 
what  groupings  should  be  made  to  obtain  a  series  as  regular  as  pos- 
sible. The  data  for  each  group  should  then  be  combined  and  -' 
select  table  graduated  bv  the  graphic  method.  These  tables  wi'' 
indicate  where  it  is  possible  to  merge  the  select  tables  into  the  ulti 
mate,  and  will  furnish  a  guide  for  drawing  the  curves  of  mortality 
for  each  of  the  early  insurance  years.  A  mathematical  formula  ap- 
proximating to  the  facts  can  generally  be  used  to  advantage  as  a 
basis,  as  then  only  the  departure  of  the  facts  from  the  formula  re- 
mains to  be  graduated  and  a  larger  scale  can  be  adopted. 

9.  Discuss  the  propriety  of  issuing  single  premium  policies  on  the 
participating  plan.   How  would  you  compute  rates  for  such  policies? 

There  does  not  appear  to  be  any  conclusive  objection  to  doing  so. 
It  is  true  that  one  considerable  item  can  be  determined  more  ac- 
curately in  advance  in  the  case  of  single  premium  than  in  the  case  of 
annual  premium  policies,  namely,  expense  of  collecting  premiums, 
also  that  the  lapse  feature  is  practically  eliminated.  There  are 
other  elements,  however,  which  are  subject  to  variation,  and  these 
are  sufficient  to  justify  issuing  the  policies  on  the  participating  plan. 


51 


As  any  lack  of  exact  justice  in  the  premium  rate  can  be  adjusted  in 
the  dividends,  the  loading  can  be  made  to  conform  to  the  general 
rule  of  the  company  for  participating  business. 

10.  Show  what  relationship  must  exist  hetzveen  the  values  of  p^ 
by  two  different  tables  of  mortality  in  order  that  the  policy  values  by 
the  two  tables  for  the  same  rate  of  interest  may  be  the  same.  Discuss 
other  forms  of  policy  as  well  as  ordinary  life. 

We  have  for  any  form  of  policy  where  it  is  the  net  premium  and  i 
the  rate  of  interest  (^jV  +  «)  (i  +  0  =  „V  +  g^^^  +  «  _  i  (Sn  —  «V) 
where  „  _iV  and  „V  are  the  terminal  reserves  for  the  {n  —  i)**  and 
n*^  years  respectively,  and  S„  the  sum  assured  in  the  w  year.  Let 
^[,1  _!_„  _i  be  the  rate  of  mortality  in  the  «'*  year  according  to  some 
other  mortality  table,  giving  the  same  terminal  reserves  but  a  dif- 
ferent  net  premium   it  .     Then 

(n_lV     +    it^)  (I    +    0   =    „V    +  ^U+n-li^n-n^) 
or,  subtracting, 
(TT^  -7t){i+t)=  (/[^i  +  „_i  -  ^[,]4.„_i)  (S„-„V). 

The  left  side  of  the  equation  is  constant  so  long  as  the  premiums 
do  not  change  so  that  the  difference  in  the  rates  of  mortality  must 
be  inversely  proportional  to  the  net  amount  at  risk.  In  the  case  of 
ordinary  life  policies  two  aggregate  tables  may  give  the  same  ter- 
minal reserves  (see  Associate  B,  question  6),  but  for  other  forms 
one  at  least  of  the  tables  must  be  in  the  analyzed  or  select  form. 
For  policies  which  have  become  paid-up  the  left  side  vanishes  and 
consequently  no  variation  in  the  rate  of  mortality  is  permissible.  For 
endowments  in  the  final  year,  the  formula  calls  for  an  infinite  differ- 
ence in  the  rates  of  mortality,  showing  an  impossible  condition. 

11.  Deduce  a  formula  for  the  distribution  of  profits  on  the  con- 
tribution plan,  and  state  what  facts  would  govern  you  in  determining 
the  constants. 

Under  what  theory  of  expense  assessment  is  a  dividend  earned 
during  the  first  policy  year? 

In  the  case  of  any  policy  we  have  („_iV  -f  it){i  +  0  —  ^-p_j_„_i 
=  (i  —  ^^_}_„_i)  „V  where  it  is  the  net  premium  ;  i  the  assumed 
rate  of  interest  and  ^^  i  „_ithe  tabular  rate  of  mortality.  Suppose 
now  the  loading  is  /  and  the  expenses  are  e,  that  the  actual  rate  of  in- 
terest earned  is/  and  the  actual  rate  of  mortality  is  ^J.^„_i  then  the 
profits  earned  during  the  year  are 

(„_iV  +  ;r  +  /-^)  (f  4-7-)-^i+n_i-(i-iri+„-l)nV 
or  substituting  from  the  equation  above 
a-e)  (I   +  ^O  +  (/-  i)  («_iV   4-  ir)   +  (^,+n_i-^+n-l) 


53 


The  rate  of  interest  earned  is  determined  by  dividing  the  interest 
earnings,  less  investment  expenses,  by  the  mean  funds  (reserve  and 
surplus)  of  the  year  less  one-half  the  interest  earnings.  An  average 
rate  should  be  used  based  on  the  company's  recent  experience.  The 
rate  of  mortality  should  be  based  on  the  company's  experience  over 
a  sufficient  period  to  give  stability.  The  expenses  after  deducting 
off-setting  items,  such  as  gain  from  lapses,  give  when  distributed  the 
.expense  factor. 

.  If  it  is  assumed  that  any  excess  of  the  cost  of  new  business  over 
the  profit  from  lapses  and  gain  from  mortality  of  recently  selected 
risks  is  properly  assessable  on  the  business  at  large  as  necessary  for 
the  maintenance  of  the  company,  a  dividend  will  be  considered  as 
earned  the  first  policy  year. 

12.  A  woman  in  receipt  of  an  income  under  an  annuity  on  her 
own  life  desires  to  have  the  annuity  changed  so  as  to  be  payable  so 
long  as  either  she  or  her  husband  survives.  What  action  would  you 
take  on  the  request  and  what  circumstances  would  you  take  into  con- 
sideration f 

In  the  case  of  such  a  request,  the  company  should  decline  to  make 
the  proposed  change  and  suggest  that  application  be  made  for  a 
survivorship  annuity,  in  favor  of  the  husband.  In  the  consideration 
of  this  application  the  various  questions  affecting  the  desirability  of 
the  risk  could  be  taken  up  in  the  regular  way.  If  it  is  desired  that 
the  annuity  be  continued  for  the  full  amount,  such  additional  pre- 
mium as  is  necessary  being  paid  now  to  the  company,  the  case  would 
be  more  favorable  for  action. 

13.  Express,  in  the  form  of  an  integral,  the  value  of  axyz\w 
and  lay  out  a  schedule  shozving  how  you  would  apply  a  formula  of 
approximate  integration  to  its  evaluation. 

See  Institute  of  Actuaries'  Text  Book,  Part  II,  Chapter  XV,  Art. 
29,  The  arrangement  of  work  given  in  Article  24,  with  the  neces- 
sary additional  column  for  the  additional  life,  would  be  convenient. 

14.  Under-average  lives  have  been  assured  with  extra  premiums 
payable  during  the  continuation  of  the  contract.  Under  what  condi- 
tions, if  at  all,  can  the  extra  premium  be  remitted^ 

Where  extra  premiums  have  been  charged  on  account  of  an  im- 
pairment they  should  not  in  general  be  remitted.  The  fact  of  the 
assured  subsequently  passing  a  good  medical  examination  is  not 
sufficient  ground  for  such  action.  If,  however,  other  circumstances 
are  such  as  to  indicate  clearly  that  an  error  was  made  by  the  medical 
examiner  originally,  the  extra  may  be  remitted.  Also  if  the  busi- 
ness has  been  transacted  under  such  conditions  that,  at  the  time  the 
application  is  made,  it  would  be  more  profitable  to  the  company  to 
remit  the  extra  and  retain  the  healthy  life  on  those  terms  than  to 
refuse  and  cause  a  lapse,  prudence  would  dictate  its  remission.  This 
could  in  general  only  occur  during  the  first  policy  year. 

15.  Show  how  Woolhouse's  formula  for  graduation  was  obtained 
and  draw  up  a  schedule  of,  operations  by  which  it  can  be  mo^t  con- 
vert en  tly  applied. 


55 


Woolhouse's  formula  for  graduation  was  derived  by  considering 
the  five  different  series,  which  can  be  formed  by  taking  values  from 
the  original  series  separated  by  quinquennial  intervals.  Each  series 
is  then  supposed  to  be  filled  out  by  central  difference  interpolation 
to  the  second  order,  and  a  final  series  formed  by  taking  the  aver- 
ages of  the  corresponding  values  in  these  five  series.  We  then  see 
that  expressing  by  u  i  the  graduated  value  we  have. 

A«x  +  6)    +   (»V«ar-3+    M   »x  +  2  "  l'?   «x  +  7  )  [ 

or  I2S  ul=  —  3  «,_7  -  2  u^_f^  +3  «^_4  +  7  u^_^  +  21 

«^_2    +   24  »^_i   +   25   «^   +   24  U^^i    -h   21    «,^.2   -f    7 

"xr3    +3«:,  +  4-2«,  +  «-3«x  +  7 

Various  methods  have  beeen  suggested  to  simplify  the  calcula- 
tions involved,  but  probably  the  simplest  is  that  of  J.  A.  Higham, 
which  consists  in  arranging  the  values  in  columnar  form,  summing 
in  threes  and  subtracting  three  times  the  result  from  ten  times  the 
central  original  value,  then  summing  the  result  in  fives  three  times 
successively  and  dividing  by  125.     See  J.  I.  A.  Vol.  31,  p.  323. 

16.  Gross  premiums  are  to  be  computed,  which,  after  covering 
certain  initial  and  renewal  expenses,  which  are  partly  proportional 
to  the  premium  and  partly  constant,  will  provide,  on  the  assumption 
of  four  per  cent  interest,  a  simple  reversionary  bonus  of  one  per 
cent  per  annum.  Give  a  formula  for  the  premium  and  state  what 
basis  of  z'aluation  should  be  adopted. 

Let  ?f  i  be  the  required  office  premium,  and  suppose  the  initial 

expenses  are  k  7t^  -\-  a  per  unit  assured  and  the  renewal  expense 

/  ^3.+^.     Also  suppose,  to  be  perfectly  general,  the  policy  in  question 

s  an  endowment,  with  a  period  of  n  years  and  with  premiums  for  m 
years.  We  have  then  where  all  monetary  values  are  on  a  four  per 
cent,  basis 

1.       ,  „  .     _,  Rx^l  —    Ra  +  nH-l  +   n(T>x  +  n  —  M^^w) 

ItxKl    +   axm-l\)  =  Axnl   +  .01    g 

+  ^TTx  +  n  +  iiix  ax  iir^Tii  +  bax  ;rrii 

Rx  +  1— Rx  +  n  4-  1  +  n{\)x^n  —  Ma;-4-n) 


Ax„l  +  tf  +  bnxn-\\ 


1 r>- 

A  net  valuation  on  a  three  and  one  half  per  cent,  basis,  with  a 
special  reservation  for  expenses  on  limited  payment  policies  after 
they  have  become  paid-up,  would  probably  on  an  average  distribu- 
tion of  the  business  be  sufficient  to  maintain  the  dividends. 


S7 


Fellow. 

17.  Policies  with  semi-annual  or  quarterly  premiums  are  usually 
valued  on  the  assumption  that  the  annual  premium  has  been  paid 
and  credit  is  allowed  in  the  assets  for  the  gross  deferred  premium, 
less  loading.  How  does  the  reserve  thus  obtained  compare  with 
that  obtained  by  valuing  the  policies  as  strictly  half-yearly  or  quar- 
terly contracts? 

See  Institute  of  Actuaries  Text  Book,  Part  II,  Chapter  XVIII, 
Articles  84  to  88. 

It  is  to  be  noted  in  this  connection  that,  in  this  country  where 
premiums  are  payable  semi-annually  or  quarterly  they  are  merely 
instalments  of  the  annual  premium  and  not  true  semi-annual  or 
quarterly  premiums ;  as  any  unpaid  balance  is  deducted  from  the 
death  claim. 

18.  (a)  What  are  the  three  principal  sources  of  surplus  and  in 
what  proportions  do  they  generally  contribute? 

(b)  How  would  you  treat  investment  expenses  in  determining 
the  share  of  profits  from  each  source? 

(a)  The  question  of  the  proportions  in  which  the  various  sources 
contribute  to  profits  is  one  to  which  various  answers  may  be  given, 
according  to  the  point  of  view.  When  we  remember,  however, 
that  what  appears  as  gain  from  surrenders  and  lapses,  is  in  large 
part  merely  a  refund  of  initial  expenses  not  yet  reimbursed  out  of 
loadings  and  that  the  bulk  of  what  appears  as  gain  from  interest  is 
really  interest  earned  on  surplus,  the  distribution  of  which  is  defer- 
red, it  would  appear  that  the  main  sources  of  profit,  at  the  present 
time,  in  the  order  of  their  importance,  are  gain  from  excess  of 
loading  over  net  expenses,  mortality  savings  and  interest  earned  in 
excess  of  the  rate  necessary  to  maintain  the  reserve.  The  two 
former  probably  contribute  about  equally,  and  the  last  considerably 
less  than  either  of  them. 

(b)  Investment  expenses  which  are  necessary  for  the  care  of 
old  investments,  including  changes  of  securities  and  reinvestments, 
should  be  considered  as  a  charge  against  interest  income. 

19.  (a)  A  company  grants,  as  surrender  values  under  its  poli- 
cies, cash,  paid-up  policies  or  extended  insurance.  How  would  you 
fix  the  relative  amounts  of  these  surrender  values? 

(b)  What  theoretical  objections  are  there  to  extended  insurance 
in  case  of  endowments? 

(a)  The  amount  of  the  paid-up  policy  to  be  given,  should  be 
determined  by  deducting  from  the  reserve  on  the  policy  a  charge 
for  initial  expenses  not  yet  reimbursed.  The  manner  of  determin- 
ing the  amount  of  this  charge  should  vary  with  the  practice  of  the 
company  regarding  dividends  and  other  features.  The  balance  after 
deducting  this  charge  should  then  be  converted  at  net  rates  into 
paid-up  assurance.  Theoretically,  a  further  charge  should  be 
made  on  account  of  the  justifiable  presumption  that  a  person  sur- 
rendering his  policy  is  a  select  life,  but,  so  far  as  paid-up  policies 
are  concerned,  this  is  already  sufficiently  provided  for,  if  the  re- 
serves used  as  a  basis  are  based,  for  premiums  as  well  as  for  other 
factors,  on  an  ultimate  table.  In  converting  into  cash,  however,  ac- 
count should  ordinarily  be  taken  of  this  possible  selection.     And  in 


59 


determining  the  period  of  extended  insurance,  a  sufficient  loading 
should  be  used  to  compensate  for  the  selection  adverse  to  the  com- 
pany, which  may  be  exercised  by  the  assured. 

(b)  Applications  are  frequently  accepted  on  the  endowment 
form  which  would  not  be  accepted  for  term  assurance,  the  expecta- 
tion being  that  near  the  end  of  the  period  the  amount  at  risk 
would  be  small  and  that  the  effect  of  extra  mortality  occurring  at 
that  time  might  be  neglected.  If  however  extended  assurance  is 
given  after  a  few  years'  premiums  have  been  paid  the  company 
would  be  on  the  risk  for  the  full  amount  as  the  reserve  would  be 
small. 

20.    Find  an  approximate  expression  for  '»^.     and  show  how  it 
must  be  modified  to  give  the  value  of  a^**^^- 

See  Institute  of  Actuaries  Text  Book,  Part  II,  Chapter  XI,  Ar- 
ticles 5  and  8. 

21.  What  are  the  legal  requirements  as  to  basis  of  valuation  in 
order  that  nezu  business  may  be  transacted? 

To  what  point  must  the  reserve  be  reduced  before  a  receivership 
can  be  insisted  on? 

In  the  State  of  New  York,  a  company  is  not  permitted  to  transact 
new  business,  if  a  statement  of  assets  and  liabilities,  including  the 
value  of  its  policies  on  the  basis  of  the  Actuaries'  or  Combined  Ex- 
perience Table  with  four  per  cent,  interest,  for  policies  issued  prior 
to  January  1st,  1901  and  on  the  basis  of  the  American  Experience 
Table,  with  three  and  a  half  per  cent,  interest,  for  policies  issued 
since  that  date  or  on  such  higher  basis  as  the  company  may  have 
adopted,  shows  an  impairment  of  more  than  fifty  per  cent,  of  the 
capital.  But  a  receiver  cannot  be  appointed  unless  the  funds  in- 
vested according  to  law,  after  deducting  the  outstanding  liabilities 
and  capital  stock,  fall  below  the  reserve  on  the  basis  of  the  Ameri- 
can Experience  Table  with  four  and  one  half  per  cent,  interest. 

See  section  82  of  the  general  Insurance  Law. 

22.  (a)  State  Gompertz's  hypothesis  as  to  the  law  of  human 
mortality  and  deduce  the  value  of  l^   in  terms  of  the  constants. 

(b)  What  modification  was  introduced  by  Makeham,  and  how 
did  it  affect  the  value  of  ixf 

See  Institute  of  Actuaries'  Text  Book,  Part  II,  Chapter  VI,  Ar- 
ticles 9,  10,  14  and  15. 

23.  (a)  Give  a  brief  outline  of  the  usual  form  of  statement  re- 
quired of  a  life  company,  enumerating  the  various  schedules. 

(b)  In  preparing  the  annual  statement  a  company  wishes  to 
write  off  two  per  cent  of  the  value  of  its  real  estate.  Show  two 
zvays  in  which  this  may  be  done. 

(a)  See  standard  blank  for  returns  of  life  companies. 

(b)  Any  desired  amount  may  be  written  off  the  value  of  real 
estate,  either  by  passing  the  amount  through  disbursements  under  a 
special  profit  and  loss  item  and  correspondingly  reducing  the  item  of 


61 


book  value  of  real  estate  under  assets,  or  by  setting  up  an  item 
in  the  liabilities  of  a  real  estate  depreciation  fund. 

24.  Give  the  formula  for  the  reserve  on  a  survivorship  annuity 
with  return  of  premiums  in  case  of  prior  death  of  the  nominee. 

What  basis  would  you  adopt  for  valuation? 

Let  (x)  be  the  assured  and  (y)  the  nominee,  and  let  7t  be  the  net 
premium  and  tt*  the  gross  premiun  per  unit;  then 


(I  +  a^)  =  ay-  a^y  +  7t\lA)ai   or   l  +  7t  = ^^f:^ 

Then  the  reserve  at  the  end  of  «  years  will  be 

—    7t  (l     +    ax+n:v  +  n)     =     (l     +    ^y  +  n)    —    (l     +    TT)    (i     +    ax+n-.y  +  n) 
I 


M : +  R 


X  +  n    y  +  n  x  +  n   y  +  n 

i'x  -r  n  -.y  ^  n 

Theoretically,  an  assurance  table  should  be  used  for  the  as- 
sured and  an  annuity  table  for  the  nominee,  but  in  practice  the 
labor  would  be  prohibitive  and  the  Carlisle  table,  besides  being  con- 
venient, provides  for  a  sufficient  reserve.  The  same  rate  of  interest 
should  be  used  as  for  other  annuities.  Three  and  one  half  per 
cent,  would  be  sufficiently  conservative. 

25.  What  will  be  the  probable  effect  upon  the  dividends  to  the 
survivors  of  an  extra  mortality  in  the  eighteenth  to  the  twentieth 
years  under  tzventy-year  endowment  policies  issued  on  the  twenty- 
year  dividend  planf    State  the  reasons  for  your  answer. 

Each  extra  death  claim  reduces  the  total  profits  to  be  divided,  by 
the  amount  of  premiums  unpaid  less  expenses  and  by  the  interest  on 
those  premiums  and  on  the  amount  of  the  claim.  If  this  amount  is 
less  than  the  share  of  profits  which  would  have  been  allotted  to  the 
policy  at  maturity,  had  the  policyholder  survived  to  that  time,  the 
reduction  in  the  number  sharing  in  the  profits  will  more  than 
counterbalance  the  reduction  in  the  total  profits  and  the  share  of 
each  survivor  will  consequently  be  increased.  In  the  case  of  twenty 
year  endowment  policies  on  the  deferred  dividend  plan  this  would 
generally  be  the  case  after  the  eighteenth  j^ear.  , 

26.  (a)  A  twenty-year  endowment  has  been  in  force  five  years 
and  it  is  desired  to  change  it  to  an  ordinary  life  policy.  What 
allozvance  should  be  made  for  the  higher  premium  paid  in  the  pastf 

(b)  What  difference  zuould  you  make  in  such  cases  as  between 
annual  dividend  and  deferred  dividend  policies? 

(a)  In  making  a  change  such  as  this  the  maximum  allowance 
which  can  be  made  is  the  difference  in  reserves  on  the  two  policies 
with  such  deduction  as  may  be  necessary  on  account  of  the  unre- 
couped   initial   expenses.       The   amount   allowed   should,   of  course, 


63 


not  exceed  the  difference  between  the  surrender  values  of  the  two 
policies,  and  a  rule  to  allow  that  difference  would  in  practice  pro- 
duce fair  results. 

(b)  The  rule  in  this  form  would  apply  to  either  annual  or  de- 
ferred dividend  policies  the  appropriate  difference  in  treatment  be- 
ing provided  for  by  the  difference,  if  any,  in  values  allowed  on  the 
two  forms  of  contract. 

27.  Evaluate  the  following  integrals  : 

(a)   /*   Ji'   e-^  dx. 

{h)   f  {x  -  af  {b  -  xf  dx. 

J  a 

^    '  J  O  X    +     1 

(a)  f    x^  e-""  dx=  3  f    x^  e-^' dx 
Jo  Jo 

'  =  6  /     X  <?-*  dx 

J  o 

=  d  f    e-=^  dx 
=  6. 

(b)  /  (x  -  ay  {b  -  xf  dx  =  5    /'%-  -  a)*  (b  -  x)  dx 

J  a  •/  a 

=  V,.  /    (.r  -  nf  dx 
J  a 

-  g\,  (^'  -   ^rf. 

(c)  /      ^ dx  =        {x  +  2)  dx  -  / dx 

^  '  J  o  .r+l  Jo  J  o   X  +  1 

=  ~  +  2a  —  loge  (i  +  a). 

28.  It  is  found  from  a  large  number  of  observations  that  the 
annual  death  rate  from  accident  among  railroad  engineers  averages 
about  five  per  thousand.  State  under  what  terms  you  would  grant 
an  engineer  a  twenty-pa^jment  life  policy. 

The  addition  of  a  little  under  .005  to  the  force  of  mortality  at 
each  age  is  equivalent,  so  far  as  annuity  values  are  concerned,  to 
adding  one  half  per  cent,  to  the  rate  of  interest.  Expressing,  then, 
net  premiums  in  terms  of  annuity  values,  we  see  that  the  extra 
mortality  is  provided  for  on  ordinary  life  policies  by  taking  the  net 
premium  at  a  rate  of  interest  one  half  per  cent,  higher  as  a  basis 


65 


and  adding  the  difference  in  the  rates  of  discount.  And  that  the 
net  premium  for  a  twenty  payment  life  policy  bears  the  same  ratio 
to  that  for  an  ordinary  life  policy  as  holds  for  the  corresponding 
premiums  for  normal  mortality  at  the  higher  rate  of  interest.  In 
other  words,  the  rule  for  a  twenty  payment  life  policy  would  be  to 
take  the  net  premium  at  a  rate  one  half  of  one  per  cent,  higher  and 
increase  it  in  the  proportion  which  the  difference  in  the  rates  of 
discount  bears  to  the  net  ordinary  life  premium  at  the  higher  rate. 
This  provides  for  an  addition  to  the  force  of  mortality  equal  to 
the  difference  in  the  force  of  discount.  Now  the  normal  rate  of 
death  from  accident  is  about  one  per  thousand  so  that  the  extra 
force  of  mortality  for  engineers,  according  to  the  data,  would  be 
about  .004  or  eighty  per  cent,  of  that  which  the  above  rule  provides 
for.  This  could  be  adjusted  by  reducing  the  extra  in  that  pro- 
portion. 


THE  FUNDAMENTAL 
PRINCIPLES  OF  PROBABILITY 


I.     The  Measurement  of  Probabilities 

As  mathematical  science  has  to  do  with  quantities  and 
their  nlations,  it  is  evident  that  before  anything  can  be 
done  in  the  way  of  developing  a  mathematical  theory  of 
probability,  we  must  establish  some  method  of  determin- 
ing the  numerical  measure  of  a  given  probability.  It  is 
true  that  without  doing  so  it  is  possible  to  lay  down 
rules  of  operation  and  deduce  their  necessary  conse- 
quences just  as  we  may  say  that  the  area  of  a  rectangle 
is  equal  to  the  product  of  its  length  by  its  breadth  even 
though  we  do  not  know  and  have  no  means  of  ascertain- 
ing either  of  those  factors.  So  also,  in  probability,  we 
are  able  to  deduce  certain  relations  among  connected 
probabilities  which  hold,  even  though  the  exact  values 
of  these  probabilities  are  unknown.  But  in  order  to  de- 
termine these  relations  it  is  necessary  to  know  the  nature 
of  the  quantities  with  which  w^e  are  dealing  and  their  re- 
lation to  the  entities  of  which  they  are  the  measure. 

Probability  then  has  to  do  w^ith  events  or  things  which, 
in  detail  and  as  to  the  individual  items,  do  not,  at  some 
specified  time,  fall  within  our  knowledge.  In  so  far  as 
anything  is  a  matter  pf  knowledge  it  is  necessarily  taken 
out  of  the  region  where  probability  holds  sway. 

Again  probability  has  to  do  with  the  frequency  with 
which  an  event  happens  or  a  specified  result  is  obtained. 


71 


Other  things  being  equal  that  event  which  happens  the_ 
more  frequentlxJs_tjie  rnore  probable.  In  fact  the  nu- 
■nTericaT  measure  which  has  been  universally  adopted  for 
^he  probability  of  an  event  under  given  circumstances  is 
the  ultimate  value,  as  the  number  of  cases  is  indefinitely" 
increased,  of  the  ratio  of  the  number  of  times  the  event 
happens  under  those  circumstances  to  the  total  possible 
number  of  times.  From  this  statement  it  follows  that  the 
greatest  possible  value  of  a  probability  is  unity  and  that 
this  expresses  the  probability  of  an  event  which  happens 
on  every  possible  occasion,  or  in  other  words,  which  is  cer- 
tain to  occur.  The  lowest  value  is  zero  which  expresses 
the  probability  of  an  event  which  never  occurs.  We 
also  see  that  those  events,  or  ways  of  happening  of  an 
event  are  equally  likely  or  have  equal  probabilities  which 
happen  in  the  long  run  in  the  same  percentage  of  the  pos- 
sible cases.  An  individual  is  said  to  be  selected  at  ran- 
dom from  a  group  when  all  the  individuals  composing 
the  group  are  equally  likely  to  be  selected. 

It  is  stated  above  that  the  measure  there  given  has  been 
universally  adopted  and  this  holds  true  in  spite  of  the  fact 
that  the  rule  has  been  stated  in  ways  which  on  their  face 
differ  widely  from  that  above  given.  The  one  most  com- 
monly given  is  that  if  an  event  can  happ£rLirua_ways_and 
fail  in  b  ways  all  of  which  are  equally  likely,  the  probability 
of  the  event  is  the  ratio  of  o^to  the  sum  of  a  and  b.  It  is 
readily  seen  that  if  we  read  mto  this  statement  the  meaning 
of  the  words  "equally  likely,"  this  measure,  so  far  as  it  goes, 
reduces  to  a  particular  case  of  that  given  above.  This  form 
of  stating  the  rule  is  useful  in  those  cases  where  the  happen- 
ing or  failure  of  the  event  can  be  analyzed  into  a  definite 
number  of  elementary  ways  of  happening,  all  of  which  we 
may,  either  on  a  priori  grounds  or  by  the  conditions  of  the 
problem,  assume  to  be  equally  likely.  For  example,  where 
three  balls  are  drawn  at  random  from  an  urn  containing 


73 


seven  white  and  three  black  balls,  all  the  different  combina- 
tions three  at  a  time  which  are  possible,  are  equally  likely 
as  we  can  go  from  any  one  combination  to  any  other  by  a 
series  of  interchanges  or  substitutions  of  one  ball  for  an- 
other each  of  which  by  the  condition  of  the  problem  that  the 
drawing  is  made  at  random  leaves  the  probability  un- 
changed. In  order  then  to  determine  the  probability  that 
the  three  balls  drawn  should  fulfill  any  given  condition,  for 
instance  should  be  all  white,  we  count  the  number  of  com- 
binations which  do  fulfill  it,  in  this  case  35,  and  the  total 
number  120,  and  the  ratio  of  these  two  numbers  gives  the 
required  probability.  It  is  for  this  reason  that  so  many 
problems  in  the  theory  of  probability  reduce  to  an  enumera- 
tion of  combinations,  permutations  or  arrangements.  But 
the  problem  in  every  case  under  all  its  various  shapes  and 
disguises  is  the  same,  namely,  to  determine  in  what  propor- 
tion of  the  possible  cases  the  event,  in  the  long  run,  tends  to 
happen.  The  nature  of  this  tendency  will  be  discussed  in 
the  note  on  repeated  trials. 

II.     The  Combination  of  Probabilities 

The  next  step  in  the  development  of  the  mathemat- 
ical theory  of  probability  is  to  determine  under  what 
conditions  the  fundamental  operations  of  addition  and 
multiplication,  with  their  inverses,  subtraction  and  divis- 
ion, give,  when  performed  between  probabilities,  inter- 
pretable  results  and  to  ascertain  the  interpretation  of 
those  results. 

In  order  that  two  probabilities  may  be  added  together, 
it  is  necessary  that  they  should  relate  to  the  same  gen- 
eral subject.  For  example,  the  result  of  adding  together 
the  chance  of  throwing  six  with  a  die  and  that  of  a  man 
aged  35  dying  within  a  year  would  not  be  capable 
of  interpretation  as  a  probability.     Again  the  two  proba- 


bilities  must  not  have  any  part  common  to  both,  as  other- 
wise that  part  would  be  included  twice  in  the  sum  and 
the  result  again  could  not  be  interpreted. 

_We  thus  see  that,  in  order  that  their  probabilities 
may  be  added  together,  two  events  must  be  mutually  ex- 
clusive, that  is,  it  must  be  impossible  for  the  two  events 
lo  happen  together.  When  this  is  the  case  the  sum  of 
the  probabilities  gives  the  chance  that  one  or  other  of 
the  events  will  happen.  This  may  be  seen  from  the  fact 
that  in  a  large  number  of  trials,  since  the  two  events 
never  happen  together,  the  number  of  times  when  either 
one  or  the  other  happens  is  found  by  adding  together  the 
number  of  times  when  they  respectively  happen.  For 
example,  consider  the  chance  of  throwing  more  than 
four  with  one  throw  of  an  ordinary  six-faced  die.  This 
can  be  done  either  by  throwing  five  or  by  throwing  six,, 
the  probability  of  each  being  one-sixth.  Then  in  the 
long  run  for  each  six  trials  five  will  be  thrown  once  and 
six  once  or  more  than  four  will  be  thrown  twice,  and  its 
probability  is  therefore  one-third. 

Again,  if  an  urn  contains  five  white,  seven  red  and 
eight  black  balls  and  one  be  drawn  at  random  the  chance 
of  draAving  a  white  one  is  five-twentieths,  that  of  draw- 
ing a  red  one  is  seven-twentieths  and  the  chance  of  draw- 
ing either  a  white  or  a  red  ball  is  the  sum  of  these  two,, 
or  twelve-twentieths,  as  may  be  readily  seen  from  the 
fact  that  twelve  of  the  twenty  balls  are  either  white  or 
red. 

From  the  above  reasoning  it  follows  since  subtraction 
is  the  inverse  of  addition  that  the  probability  to  be  sub- 
tracted must  be  included  in  and  form  a  part  of  the  prob- 
ability from  which  it  is  to  be  subtracted.  For  example, 
in  the  case  above  cited,  if  we  know  that  the  probability 
of  either  a  white  or  red  ball  being  drawn  is  three-fifths 
and  that  the  probability  of  a  white  ball  being  drawn  is 


77 


one-fourth,  then,  by  subtraction,  the  probability  of  a  red 
ball  is  the  difference  or  seven-twentieths. 

A  prominent  example  of  the  subtraction  of  probabilities 
arises  in  connection  with  the  fact  that  since  an  event  must 
either  happen  or  fail  the  chance  of  one  or  the  other  result 
is  unity.  So  that  by  subtracting  from  unity  the  probability 
of  an  event  happening  we  get  the  probability  of  its  failing. 
These  two  probabilities  are  said  to  be  complementary  to  one 


another. 

Since  a  probability  is  a  fraction  of  which  the  denom- 
inator is  the  number  of  times  an  event  can  possibly  hap- 
pen, and  the  numerator  is  the  number  of  times  it  does 
happen,  it  follows  that  if  the  product  of  two  probabilities 
is  to  be  itself  a  probability  they  must  be  so  connected 
that  the  denominator  of  one  represents  the  same  class 
of  events  as  the  numerator  of  the  other.  In  other  words, 
if  one  factor  is  the  probability  of  an  event  happening, 
the  second  factor  must  be  the  probability  of  some  other 
event  also  happening  if  the  first  does,  and  in  that  case 
the  product  represents  the  probability  of  both  events  hap- 
pening together.  For  example,  suppose  an  urn  contains 
one  hundred  balls  of  which  twenty  are  ivory  and  that,  of 
these  twenty,  five  are  white.  Then,  if  one  ball  be  drawn 
at  random  the  chance  that  it  is  ivory  is  one-fifth,  and  the 
chance  that  if  it  is  ivor\^  it  will  be  white  is  one-fourth,  and 
we  see,  from  the  fact  that  of  the  total  of  one  hundred 
balls  five  are  white  ivory  ones,  that  the  probability  of  a 
white  ivory  ball  being  drawn  is  one-twentieth  which  is  the 
product  of  one-fourth  and  one-fifth.  Similarly,  if  at  a 
given  age  one  person,  on  an  average,  in  each  125  dies  within 
one  year  and  of  those  who  die  at  that  age  one  in  eight  on 
an  average  die  from  accident,  then  in  the  long  run  out  of 
each  thousand  persons  eight  will  die  within  the  year  and 
of  these  eight  one  will  die  from  accident.  Or  the  proba- 
bility of  a  person  dying  from  accident  within  one  year  is 


79 


one  one-thousandth  or  the  product  of  one  hundred  and 
twenty-fifth  and  one-eighth. 

We  have  said  that  one  of  the  factors  is  the  probability 
that  if  one  event  happens  another  event  will  also  happen. 
This  probability  may  be  different  from  the  probability 
that  if  the  first  event  fails  the  second  will  happen  or  it 
may  be  the  same.  In  the  former  case,  the  events  are 
said  to  be  dependent  and  it  is  necessary  to  attend  to  the 
distinction  above  noted.  In  the  latter  case,  the  events 
are  said  to  be  independent  and,  as  the  probability  to  be 
used  in  the  multiplication  is,  in  this  case,  equal  to  the 
general  probability  that  the  second  event  shall  happen,  it 
is  unnecessary  to  attend  to  the  distinction.  This  gives  the 
rule  that  to  find  the  chance  that  both  of  two  independent 
events  shall  happen,  we  multiply  together  their  respective 
■ probabilities. 

The  rule  for  independent  events  is  the  one  which  is  most 
frequently  used  in  practice,  because  in  a  great  many  cases, 
we  may,  in  the  absence  of  any  reason  to  doubt  it,  assume 
that  the  two  events  are  independent.  This,  however,  does 
not  alter  the  fact  that  where  two  probabilities  are  multi- 
plied together  to  give  a  resultant  probability  one  of  the 
factors  is  always,  either  expressly  or  by  implication,  the 
probability  that  if  one  event  happens  another  will  happen 
also. 

Conversely  the  probability  of  a  compound  event  may  be 
divided  by  the  probabilitv-  of  one  of  its  component  simple 
events,  the  happening  of  which  is  necessary  to  that  of  the 
compound  event,  and  the  quotient  gives  the  probability  that 
if  the  simple  event  happens  the  remainder  of  the  com- 
pound event  will  also  happen.  Or  this  latter  probability 
may  be  known  and  be  used  as  the  divisor  and  the  quotient 
would  give  the  probability  of  the  simple  event.  For  ex- 
ample, suppose  we  know  that  at  a  given  age  the  prob- 
ability of  death  from  accident  within  one  year  is  one  per 


81 


thousand  and  that  of  the  total  deaths  at  that  age  one  in 
ten  is  by  accident,  then  by  division  we  find  the  total  chance 
of  death  within  one  year  to  be  ten  per  thousand. 

All  the  computations  of  probabilities  are  made  by  the 
application  direct  or  indirect  of  these  four  rules,  and  most 
of  the  errors  that  arise  in  such  computations  come  from 
neglect  to  observe  the  restrictions  and  conditions  under 
which  these  various  operations  are  permissible.  The  fol- 
lowing is  a  good  example  of  the  combination  in  one  com- 
putation of  the  various  elementarj^  rules. 

A  number  of  urns  are  filled  with  balls.  In  each  of  one-  / 
tenth  of  the  total  number  of  urns  nine-tenths  of  the  balls 
are  white  and  in  each  of  the  remainder  four-tenths  are 
white.  From  an  urn  taken  at  random  a  ball  is  drawn  at 
random,  found  to  be  white  and  returned.  What  is  the 
chance  that  a  second  raadom  drawing  from  the  same  urn 
would  give  a  white  ball  ? 

Here  the  a  priori  probability  of  selecting  an  urn  in  which 
nine-tenths  of  the  balls  are  white  is  one-tenth  and,  conse- 
quently the  chance  that  such  an  um  will  be  selected  and 
a  white  ball  drawn  is  by  multiplication  nine  one-hundredths. 
Also  the  chance  of  selecting  one  of  the  others  is  the  com- 
plement of  one-tenth  or  nine-tenths,  and  consequently  the 
chance  that  such  an  um  will  be  selected  and  a  white  ball 
drawn  is,  by  multiplication,  thirty-six  one  hundredths. 
Hence,  by  addition  the  total  chance  of  a  white  ball  beings 
drawn  is  forty-five  one  hundredths.  From  this  we  see  by 
division  that  if  a  white  ball  is  drawn  the  chance  that  it  is 
drawn  from  one  of  the  first  set  of  urns  is  nine  forty-fifths 
or  one-fifth  and  the  chance  that  it  is  drawn  from  one  of  the 
second  set  is  four-fifths.  If  then  a  second  drawing  is  made 
from  the  same  urn  the  chance  that  the  urn  will  belong  to 
the  first  set  and  that  a  white  ball  will  be  drawn  is  nine- 
tenths  of  one-fifth  or  nine-fiftieths,  and  the  chance  that  the 
urn  will  belong  to  the  second  set  and  that  a  white  ball  will 


83 


be  drawn  is  four-tenths  of  four-fifths  or  sixteen-fiftieths. 
Hence  the  total  probability  of  a  white  ball  being  drawn 
is  twenty-five  fiftieths  or  one-half. 

III.     Expectation  or  Mean  Values 

Frequently  in  applications  of  the  theory  of  probability 
the  important  element  in  the  problem  is  the  value  assumed, 
in  the  various  contingencies,  by  some  variable  quantity. 
This  quantity  may  be  the  number  of  successes  in  a  given 
number  of  trials,  the  distance  of  a  point  from  another 
point,  a  line  or  a  plane,  the  duration  of  life,  the  amount  of 
money  to  be  paid  or  received  or  the  present  value  of  such 
amount  or  any  other  quantity  connected  with  the  result 
of  the  trial,  including  any  function  explicit  or  implicit  of 
any  of  these  quantities.  In  these  cases  the  special  point  to 
which  attention  is  generally  directed  is  the  mean  value  of 
the  variable  quantity,  or  the  ultimate  value,  as  the  number 
of  trials  is  indefinitely  increased,  of  the  average  of  the 
values  of  the  variable  resulting  from  the  various  trials. 
In  certain  cases  this  function  is  also  called  the  expectation 
from  each  trial,  the  idea  being  that  a  certain  aggregate 
result  is  expected  from  a  large  ntimber  of  trials,  and  that, 
by  dividing  this  aggregate  result  by  the  number  of  trials, 
we  get  the  expectation  from  each.  It  is  evident  that  this 
average  will  be  determined  by  multiplying  each  value  of 
the  variable  by  the  number  of  times  it  occurs  and  divid- 
ing the  sum  of  the  products  by  the  total  number  of  trials. 
In  the  ultimate  limit  this  is  equivalent,  from  the  definition 
of  probabilities,  to  multiplying  each  value  of  the  variable  by 
the  probability  of  its  occurrence  and  adding  all  the  pro-i 
ducts  together.  For  example,  suppose  one  dollar  is  to  be 
received  if  a  coin,  about  to  be  tossed,  should  turn  up  head. 
In  a  large  number  of  such  trials  the  net  results  would  be 
that  one  dollar  would  be  received  for  every  two  trials  or 


85 


the  expectation  from  one  trial  is  one-half  of  one  dollar,  or 
one  dollar  multiplied  by  the  chance  of  head  being  thrown. 

It  follows  from  the  way  in  which  the  mean  value  is  ar- 
rived at,  that  the  mean  value  of  the  sum  of  the  values  of 
the  variable  resulting  from  a  specified  number  of  trials  or 
the  expectation  from  such  trials  may  be  obtained  by  mul- 
tiplying the  expectation  from  each  single  trial  by  the  num- 
ber of  trials.  Thus  the  expectation  of  success  in  one  trial 
is  equal  to  the  probability  and  the  expected  number  of  suc- 
cesses in  a  given  number  of  trials  is  the  product  of  that 
number  into  the  probability. 

An  important  consideration  arises  in  connection  with  the 
mean  value  of  the  product  of  two  independent  variables, 
that  is,  of  the  product  of  two  quantities  whose  variations 
depend  on  contingencies  which  are  independent  of  each 
other.  In  the  computation  of  this  mean  value  all  the  pos- 
sible values  of  the  product  and  their  respective  probab- 
ilities will  be  represented  by  the  various  terms  in  the  ex- 
pression for  the  product  of  the  mean  values  of  the  indi- 
vidual variables,  so  that  the  mean  value  of  the  product  is 
equal  to  the  product  of  the  mean  values.  As  an  example, 
suppose  a  sum  of  money  is  to  be  received  which  is  to  be 
determined,  by  tossing  a  penny  to  decide  whether  payment 
will  be  in  dimes  or  in  nickels,  and  by  throwing  a  die  to 
determine  the  number  of  coins,  the  number  to  be  equal 
to  the  number  thrown.  Then  the  mean  value  of  each  coin 
is  seven  and  one-half  cents,  and  the  mean  number  of  coins 
is  three  and  one-half  and  the  expectation  therefore  twenty- 
six  and  a  quarter  cents,  as  may  also  be  seen  by  directly 
taking  the  mean  of  the  amounts  payable  in  the  twelve 
equally  likely  possible  cases. 

It  is  to  be  carefully  noted  that  this  rule,  that  the  mean 
value  of  the  product  is  equal  to  the  product  of  the  mean 
values,  applies  only  when  the  two  quantities  involved  are 
entirely  independent     Where  the  variation  of  one  quantity 


87 


is  in  any  way  dependant  on  that  of  the  other,  this  relation 
cannot  be  depended  upon.  As  a  case  in  point,  the  mean 
value  of  the  square  of  a  variable  quantity  is  always  greater 
than  the  square  of  the  mean  value.  This  appears  from 
general  considerations,  but  it  may  be  also  demonstrated 
algebraically  and  the  difference  may  be  shown  to  be  equal 
to  the  mean  value  of  the  square  of  the  difference  between 
the  actual  value  of  the  variable  and  its  mean  value.  For 
example,  in  one  trial  the  mean  value  of  the  square  of  the 
number  of  successes  is  equal  to  the  probability  of  success 
since  it  is  equal  to  unity  multiplied  by  that  probability 
added  to  zero,  multiplied  by  the  complementary  probability. 
We  have  already  seen  that  the  mean  value  of  the  number  of 
successes  is  also  equal  to  the  same  probability.  The  mean 
value  of  the  square  of  the  departure  of  the  actual  from  the 
expected  is  seen  by  a  direct  computation  to  be  equal  to  the 
product  of  the  probability  into  its  complement,  or  to  the 
difference  between  the  probability  and  its  square,  which  is 
in  agreement  with  the  statement  made  above. 

IV.    Repeated  Trials 

Where  a  number  of  trials  are  made  the  chance  that  the 
event  should  happen  every  time  is,  by  the  principle  of  the 
multiplication  of  probabilities,  equal  to  the  probability  of 
success  at  a  single  trial  raised  to  a  power  equal  to  the  num- 
ber of  trials.  Similarly  where  p  is  the  probability  of  suc- 
cess at  a  single  trial  and  q  the  complementary  probability 
of  failure,  the  chance  that  of  n  trials,  r  particular  ones 
should  be  successful  and  the  remainder  unsuccessful,  is  de- 
termined by  multiplying  p  raised  to  the  rth  power,  by  q 
raised  to  a  power  equal  to  the  difference  between  n  and  r. 
We  thus  see  that  the  probability  of  the  success  at  any  par- 
ticular r  trials  only,  is  equal  to  the  probability  for  any  other 
particular  r  trials  only,  and  that  consequently  the  total  prob- 


89 


ability  of  exactly  r  successes  is  found  by  multiplying  this 
probability  by  the  number  of  combinations  of  n  things  r  at 
a  time.  Accordingly  the  ratio  of  the  chance  of  exactly  r 
successes  to  that  of  exactly  r—  1  is  compounded  of  the  ratio 
of  the  chance  of  success  on  a  particular  r  occasions  only  ta 
that  of  success  on  a  particular  r  -  1  occasions  only,  which 
is  equal  to  the  ratio  of  p  to  q,  and  the  ratio  of  the  number 
of  combinations  of  n  things  r  at  a  time  to  the  number  r  - 1 
at  a  time,  which  is  equal  to  the  ratio  of  « -|-  1  -  r  to  r.  This 
ratio  will  be  greater  than  unity  or  the  chance  of  exactly  r 
successes  will  be  greater  than  that  of  exactly  r  - 1,  if  p 
(n+1  -^)  is  greater  than  ^r  or  if  r  is  less  than  p  (n-f-1). 
We  thus  see  that  the  most  probable  number  of  successes  is 
defermtned  hy  taking  the  integral  part  of  the  expected  num- 
ber of  successes  in  n  -f-  1  trials  and  that  the  probability  in- 
creases regularly,  without  any  retrograde  movement  up  to 
the  maximum  and  then  decreases  in  the  same  way. 

We  may  have  r  successes  out  of  n  -|-  1  trials  in  two  ways 
only,  by  r  -  1  successes  in  n  trials  followed  by  success  in  the 
other  or  by  r  successes  followed  by  failure,  its  probability 
therefore  is  intermediate  in  value  between  those  of  r  -  1  and 
r  successes  in  n  trials.  We  thus  see  that  the  probability  of 
the  most  probable  number  of  successes  in  w  -  1  trials  is  in 
general  less  than  the  corresponding  probability  for  n  trials 
as  it  must  be  less  than  the  greater  of  the  probabilities  of 
some  two  consecutive  numbers  of  successes  unless  their  two 
probabilities  are  equal,  in  which  case  it  will  be  equal  to 
either. 

We  thus  see  that  as  the  number  of  trials  increases  the 
maximum  value  of  the  probability  of  an  exact  number  of 
successes  tends  to  diminish  sometimes  halting  but  never 
retrograding.  It  follows  that  the  chance,  that  the  actual 
number  of  successes  will  be  within  any  definite  number  of 
the  expected,  decreases  as  the  number  of  trials  is  increased. 
This  result  appears  at  first  sight  inconsistent  with  the  funda- 


91 


mental  definition  of  the  measure  of  probability,  but  it  is  not 
in  fact  so  inconsistent  as  the  definition  refers  to  the  ratio  of 
the  number  of  successes  to  the  number  of  trials  and  this  re- 
lates only  to  the  number  of  successes.  We  shall  proceed  to 
investigate  the  variations  of  the  ratio  of  successes. 

We  have  seen  in  the  note  on  mean  values  that  the  mean 
value  of  the  square  of  the  departure  of  the  actual  from  the 
expected  number  of  successes  in  one  trial  is  equal  to  p  q. 
Also  the  departure  of  the  actual  from  the  expected  in  n 
trials  is  equal  to  the  algebraic  sum  of  the  departure  in  the 
individual  trials,  so  that  the  square  of  the  departure  in  n 
trials  is  equal  to  the  sum  of  the  squares  of  the  departures 
in  the  individual  trials  together  with  twice  the  sum  of  their 
products  two  and  two.  But  these  departures  for  the  in- 
dividual trials  are  independent  of  one  another  and  their 
mean  values  are  each  zero,  consequently  by  the  principle 
established  in  the  note  on  Mean  Values,  so  also  is  the  mean 
value  if  the  product  of  any  pair  of  them.  We  thus  see  that 
the  mean  value  of  the  square  of  the  departure  of  the  actual 
number  of  successes  from  the  expected  number  in  n  trials 
is  n  times  the  value  for  one  trial  or  is  equal  to  «  p  q.  But 
the  departure  of  the  actual  proportion  of  successes  from 
the  expected  is  found  by  dividing  the  departure  of  the  num- 
ber by  n  the  number  of  trials,  so  that  the  mean  value  of  its 
square  is  determined  by  dividing  n  p  q  by  the  square  of  n 
or  by  dividing  p  q  by  n.  This  quotient  is  seen  to  diminish 
as  n  increases. 

Let  now  h  be  an  assigned  departure  and  let  P  be  the 
probability  that  the  actual  proportion  of  successes  differs 
from  the  expected  by  more  than  h.  Then  it  is  evident  that 
the  mean  value  of  the  square  of  the  departure  is  not  less 
than  P  times  the  square  of  h.  Or  in  other  words  n  P  is 
not  greater  than  the  quotient  of  p  q  by  the  square  of  h. 
It  follows  then  that  no  matter  how  small  h  is,  provided  it 
does  not  vanish,  we  can  by  taking  n  large  enough  make  P 


93 


less  than  any  assigned  quantity.  We  thus  see  that  as  n  in- 
creases the  actual  proportion  becomes  more  and  more  nearly 
certain  to  agree  within  narrower  and  still  narrower  limits 
with  the  expected  proportion.  This  is  the  nature  of  the 
tendency  to  the  ultimate  value  as  the  number  of  trials  is  in- 
definitely increased. 


OF  THE 

UNIVERSITY 

OF 


9» 


OS  T«5  I^^"^ 


DATE 

OVERDUE. 


YC  23597 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


